Minimal-rank HSS-representation with diagonal coupling matrices and sparse transfer matrices for solving electrically-large integral equations

We develop a method to generate an efficient HSS (Hierarchically Semiseparable) representation of the dense integral operators. In this representation, we ensure the rank of each admissible block to be the minimal one required by accuracy. More important, we successfully diagonalize the coupling matrices, and also make the transfer matrices sparse. As compared to H²- or HSS-representations whose coupling and transfer matrices are dense, this new representation is much more efficient. Our theoretical analysis shows a strict O(N) complexity for storage and matrix-vector multiplication in both surface and volume integral equation (IE) solvers. Numerical experiments have been carried out on 3-D volume IEs for scattering analysis, which have validated the proposed new representation.